Dynamical Classification of some Birational Maps of C2

Abstract

This dissertation addresses three different problems in the study of discrete dynamical systems. Firstly, this work dynamically classifies a 9−parametric family of planar birational maps f : C2 → C2 that is f(x, y) =

α0 + α1x + α2y, β0 + β1x + β2y γ0 + γ1x + γ2y

, where the parameters are complex numbers. This is done by finding the dynamical degree δ for the degenerate and non degenerate cases of F that is the extended map of f in projective space. The dynamical degree δ defined as δ(F) := lim n→∞ (deg(Fn)) 1 n , indicates the subfamilies which are chaotic, that is when δ > 1, and otherwise. The study of the sequence of degrees dn of F shows the degree growth rate of all the subfamilies of f. This gives the families which have bounded growth , or they grow linearly, quadratically or grow exponentially. The family f includes the birational maps studied by Bedford and Kim in [18] as one of its subfamily. The second problem includes the study of the subfamilies of f with zero entropy that is for δ = 1. These includes the families with bounded (in particular periodic), linear or quadratic growth rate. Two transverse fibrations are found for the families with bounded growth. In the periodic case the period of the families is indicated. It is observed that there exist infinite periodic subfamilies of f, depending on the parameter region. The families with linear growth rate preserve rational fibration and the quadratic growth rate families preserve elliptic fibration that is unique depending on the parameters. In all the cases with zero entropy all the mappings are found up to affine conjugacy. Thirdly, it deals with non-autonomous Lyness type recurrences of the form xn+2 = an + xn+1 xn , where {an}n is a k-periodic sequence of complex numbers with minimal period k. We treat such non-autonomous recurrences via the autonomous dynamical system generated by the birational mapping Fak ◦ Fak−1 ◦ · · · ◦ Fa1 where Fa is defined by Fa(x, y) = (y, a + y x ). For the cases k ∈ {1, 2, 3, 6} the corresponding mappings have a rational first integral. By calculating the dynamical degree we show that for k = 4 and for k = 5 generically the dynamical system is no longer rationally integrable. We also prove that the only values of k for which the corresponding dynamical system is rationally integrable for all the values of the involved parameters, are k ∈ {1, 2, 3, 6}.